I'm working out of Baby Rudin, chapter 1 problem 7. The problem is "Fix b >1, y > 0, and prove that there is a unique real x such that by completing the following outline..."

I'm at part f) "Let A be the set of all w such that , and show that satisfies .

What I'm wondering about - if m is a positive integer, since b > 1, is it a leap to assume that there's an integer k such that ?

You can easily enough justify that "leap" by writing b = 1 + h, where h > 0. Then (since the right-hand side is just the first two terms of the binomial expansion), and that can clearly be made arbitrarily large.